Fault Tolerant Computing CS 530DL

Transcription

1 Fault Tolerant Computing CS 530DL Additional Lecture Notes Modeling Yashwant K. Malaiya Colorado State University March 8,

2 Quantitative models Derived from first principles: Arguments are actual things measured Example: monthly mortgage payment rp c = 1 (1 + r) N Empirical Arguments are just parameters Example Ideal body weight in kg = a+b (height in in. - c) Men: a=52 kg, b= 1.9, c = 60 in. Combined Empirical with interpretation of parameters From first principles, adjusted to fit March 8,

3 Derived using first principles Obtain model Understand how things work Use approximations Derive formulas Validate: Get real data See if it fits. If not, Make adjustments Get alternative models Use the model Plug in values to estimate March 8,

4 Empirical models Look at data See if it resembles a function Linear, quadratic, logarithmic, exponential.. Involving 1, 2 or more parameters See if it fits If not try something more complex If it fits, see if an interpretation of the parameters is possibile Not necessary, but will be good. March 8,

5 Curve fitting Get x-axis and y-axis numbers (x i, y i ). Draw a scatter plot. Fit using Excel: linest( ) or logest( ) functions Select plot, rc, add trendline, select display options Use Solver for general fitting March 8,

6 Branch Cov Defects Example Defects y = x R² = Defects Linear (Defects) March 8,

7 Curve fitting vs Predictive Capability A good model has good predictive capabilities. Curve fitting partial data may not necessarily identify the best model. March 8,

8 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Chapter 5

9 Curve Fitting Often, we have data points and we want to find an equation that fits the data Simplest equation is that of a straight line

10 Curve Fitting Example A spring is placed between two flat plates and force is slowly applied to the upper plate, causing the spring to compress. When the force reaches a pre-load value of 25 N, the location of the upper plate is recorded. As the upper plate continues to move, the distance that the top plate moves, d, and the force required to move the plate, F, are recorded. As soon as the upper plate has moved 10 mm, the test is ended. F

11 Curve Fitting Example Data from the test is shown here. The slope of the load-displacement curve is called the spring constant.

12 Curve Fitting Example Data from a test how do we find the slope? We could draw in a line by hand and estimate its slope by estimating the coordinates of two points

13 Curve Fitting Example How do we find the line that best fits the data? Find the equation of the line that minimizes the sum of the squares of the differences between the values predicted from the equation and the actual data values Why do we square the differences? Because we are interested in the magnitudes of the differences. If one point is above the line and other is below it, we don t want these difference to cancel each other

14 Curve Fit Parameters In Excel, we can choose to have the equation of the bestfit line (the trendline) displayed in the form where and m = the slope of the line b = the intercept of the line with the y-axis

15 Equation of Trendline In this example, the slope of the line (the stiffness of the spring) equals 72.3 N/mm When the deflection (x) equals zero, the force (y) equals 23.1 pounds. This is consistent with our nominal pre-load of 25 pounds

16 Correlation Coefficient The correlation coefficient (R 2 ) is a measure of how well the trendline fits the data A value of one represents a perfect fit In our example, the line fit is very good

17 Curve Fitting Example Consider these five data points: x y

18 Linear Curve Fit Poor Fit

19 Try Second Order Polynomial Fit Better, but still not very good

20 Third-Order Polynomial Fit Perfect Fit! These point were calculated from the equation:

21 Correlation Coefficient A good fit to data is relative In the case of the spring example, the data should fit a mathematical model, and so an R 2 value of close to one is expected For other cases, a much lower R 2 value is expected Consider a comparison of final exam scores in a class vs. homework averages We would expect that students who do well on HW will generally do well on the final exam, but there will be exceptions

22 Exam vs. HW Example Note the presence of outliers data points that don t fit the trend However, there is clearly a trend in the data

23 Exponential and Power Equations Consider this equation: Here is a graph of the equation:

24 Exponential Equation If we plot the ln(y) instead of y, then we have the equation of a straight line

25 Exponential Equation This plot is not particularly useful, since it requires us to read the ln of the dependent variable y, instead of y itself

26 Exponential Equation A better way is to display the y values on a logarithmic scale:

27 Exponential Equation We call this a semi-log plot since one of the axes is logarithmic Note that we have used a base-10 scale: remember that a logarithm can be converted to a logarithm of another base by multiplying by a constant: Therefore, the equation will produce a straight line on semi-log axes regardless of the base of the logarithmic scale. Base 10 normally used easiest to read Logarithmic scales can be specified in Excel and MATLAB

28 Power Equation Consider the equation for the volume of a sphere: Plot:

29 Power Equations If we plot the log(y) and log(x), then we have the equation of a straight line

30 Power Equations This is called a log-log plot, since both axes are logarithmic